The alternate formulation is closely related to the following fundamental definition from Ramsey Theory

Definition: Let $\phi : 2^X \to \text{Bool}$ be a property pertaining to subsets of the set $X$. The property $\phi$ is called partition regular if, for every partition $$X = X_1 \uplus X_2 \dots \uplus X_n $$ we have $\phi(X_i)$ for at least one $i$.

Clearly, every ultrafilter corresponds to a partition regular property, $\phi(Y) = Y\in\mathcal U$. In the other direction, it is a reasonably easy exercise to show that every partition regular property is given by a collection of ultrafilters $\phi(Y) = \bigvee \lbrace Y \in \mathcal U : \mathcal U\rbrace$. See for example theorem 3.11 in Hindman & Strauss "Algebra in the Stone-Čech compactification".

That said, I've never seen the formulation with fixed $n$, like $n=3$, before.

The alternate formulation is closely related to the following fundamental definition from Ramsey Theory

Definition: Let $\phi : 2^X \to \lbrace\text{true},\text{false}\rbrace$ be a property pertaining to subsets of the set $X$. The property $\phi$ is called partition regular if, for every partition $$X = X_1 \uplus X_2 \dots \uplus X_n $$ we have $\phi(X_i)$ for at least one $i$.

Clearly, every ultrafilter corresponds to a partition regular property, $\phi(Y) = Y\in\mathcal U$. In the other direction, it is a reasonably easy exercise to show that every partition regular property is given by a collection of ultrafilters $\phi(Y) = \bigvee \lbrace Y \in \mathcal U : \mathcal U\rbrace$. See for example theorem 3.11 in Hindman & Strauss "Algebra in the Stone-Čech compactification".

That said, I've never seen the formulation with fixed $n$, like $n=3$, before.